Mathematical Sciences: Investigations in Spectral & Fractal Geometry: Vibrations of Fractal Drums, Spectral Zeta Functions, Analysis on Fractals, & Variational Ellip

数学科学：光谱研究

• 批准号: 9207098
• 负责人: Michel Lapidus
• 金额: $ 9.09万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1996-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9207098&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Investigations; Spectral; &amp

Lapidus will investigate sharpened forms of the asymptotic formula for the eigenvalue distribution of Laplacians with various boundary conditions. The goal is to make yet clearer the role of the Minkowski dimension of the boundary in this distribution. There are analogous formulas for the trace of the associated heat semigroup. The case in which not only the boundary of the domain, but the domain itself, is fractal will be studied. The relationship between the geometry of a manifold and the invariants of the elliptic differential operators on it is one of the main unifying themes of contemporary mathematics. The case of a region with fractal boundary, so irregular that its dimension is no longer an integer, has been the setting for some recent breakthroughs relating spectra information to geometry. This work has applications to the study of porous media and the scattering of waves from fractal surfaces.

Lapidus将研究具有各种边界条件的拉普拉斯特征值分布的渐近公式的简化形式。我们的目标是使边界的闵可夫斯基维在这个分布中的作用更加清晰。对于伴生热半群的轨迹，有类似的公式。在这种情况下，不仅领域的边界，而且领域本身，是分形的将被研究。流形的几何形状与其上的椭圆微分算子的不变量之间的关系是当代数学的主要统一主题之一。具有分形边界的区域如此不规则，以至于它的维度不再是整数，这种情况已经成为最近将光谱信息与几何联系起来的一些突破的背景。这项工作在多孔介质和分形表面波的散射研究中具有应用价值。